Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [17,14,Mod(1,17)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(17, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("17.1");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 17 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 17.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(18.2292579218\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 46930 x^{6} + 462632 x^{5} + 679652608 x^{4} - 13193888768 x^{3} + \cdots - 517210568441856 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{11}\cdot 3^{3} \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{8} - x^{7} - 46930 x^{6} + 462632 x^{5} + 679652608 x^{4} - 13193888768 x^{3} + \cdots - 517210568441856 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 13\nu - 11734 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 23159 \nu^{7} + 373173 \nu^{6} - 1030880186 \nu^{5} - 9478037200 \nu^{4} + \cdots + 60\!\cdots\!68 ) / 192265058304000 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 438677 \nu^{7} - 46696239 \nu^{6} + 24850833998 \nu^{5} + 1641821889520 \nu^{4} + \cdots - 93\!\cdots\!64 ) / 961325291520000 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 796051 \nu^{7} - 15081257 \nu^{6} + 36790182274 \nu^{5} + 148742478160 \nu^{4} + \cdots - 38\!\cdots\!32 ) / 320441763840000 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 289453 \nu^{7} + 10343031 \nu^{6} - 12492565342 \nu^{5} - 270192531440 \nu^{4} + \cdots + 88\!\cdots\!76 ) / 96132529152000 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 690797 \nu^{7} - 1173399 \nu^{6} + 33523347518 \nu^{5} - 120264305360 \nu^{4} + \cdots - 32\!\cdots\!64 ) / 192265058304000 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 13\beta _1 + 11734 \)
|
| \(\nu^{3}\) | \(=\) |
\( -7\beta_{7} + 10\beta_{6} + 9\beta_{5} + 4\beta_{4} - 258\beta_{3} + 2\beta_{2} + 17981\beta _1 - 158230 \)
|
| \(\nu^{4}\) | \(=\) |
\( 301 \beta_{7} - 150 \beta_{6} - 1411 \beta_{5} + 404 \beta_{4} - 14842 \beta_{3} + 22432 \beta_{2} + \cdots + 210606118 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 196577 \beta_{7} + 350046 \beta_{6} + 304015 \beta_{5} + 202780 \beta_{4} - 7575486 \beta_{3} + \cdots - 4419138966 \)
|
| \(\nu^{6}\) | \(=\) |
\( 15311585 \beta_{7} - 2241598 \beta_{6} - 50584207 \beta_{5} + 398308 \beta_{4} - 528984322 \beta_{3} + \cdots + 4169684661510 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 4729050937 \beta_{7} + 9635314510 \beta_{6} + 8441329847 \beta_{5} + 6816872892 \beta_{4} + \cdots - 111071953178870 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−156.869 | 2039.83 | 16415.7 | −14529.3 | −319985. | −291334. | −1.29005e6 | 2.56657e6 | 2.27919e6 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −125.118 | 179.161 | 7462.64 | −26752.0 | −22416.3 | 107164. | 91256.6 | −1.56222e6 | 3.34717e6 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −94.4497 | −2226.03 | 728.753 | −6162.27 | 210248. | −51069.2 | 704902. | 3.36089e6 | 582025. | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −27.2131 | 1458.98 | −7451.45 | 30637.0 | −39703.4 | −443175. | 425707. | 534296. | −833728. | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −17.3728 | −815.982 | −7890.18 | 26529.8 | 14175.9 | 510480. | 279393. | −928497. | −460897. | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 84.9807 | 780.675 | −970.289 | −25720.9 | 66342.2 | 52712.7 | −778617. | −984870. | −2.18578e6 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 125.277 | −2079.44 | 7502.41 | 49469.9 | −260507. | −51860.0 | −86390.3 | 2.72976e6 | 6.19746e6 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 145.765 | −665.185 | 13055.4 | −18160.2 | −96960.6 | −532495. | 708909. | −1.15185e6 | −2.64711e6 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(17\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 17.14.a.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 153.14.a.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 17.14.a.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 153.14.a.c | 8 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 65 T_{2}^{7} - 45082 T_{2}^{6} - 2685256 T_{2}^{5} + 616399168 T_{2}^{4} + \cdots - 13\!\cdots\!48 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(17))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + \cdots - 13\!\cdots\!48 \)
|
| $3$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
|
| $5$ |
\( T^{8} + \cdots - 44\!\cdots\!00 \)
|
| $7$ |
\( T^{8} + \cdots - 52\!\cdots\!44 \)
|
| $11$ |
\( T^{8} + \cdots + 67\!\cdots\!00 \)
|
| $13$ |
\( T^{8} + \cdots + 10\!\cdots\!52 \)
|
| $17$ |
\( (T + 24137569)^{8} \)
|
| $19$ |
\( T^{8} + \cdots + 44\!\cdots\!00 \)
|
| $23$ |
\( T^{8} + \cdots - 58\!\cdots\!40 \)
|
| $29$ |
\( T^{8} + \cdots - 94\!\cdots\!00 \)
|
| $31$ |
\( T^{8} + \cdots + 47\!\cdots\!40 \)
|
| $37$ |
\( T^{8} + \cdots - 36\!\cdots\!84 \)
|
| $41$ |
\( T^{8} + \cdots + 20\!\cdots\!00 \)
|
| $43$ |
\( T^{8} + \cdots - 63\!\cdots\!76 \)
|
| $47$ |
\( T^{8} + \cdots - 62\!\cdots\!00 \)
|
| $53$ |
\( T^{8} + \cdots + 73\!\cdots\!20 \)
|
| $59$ |
\( T^{8} + \cdots + 99\!\cdots\!20 \)
|
| $61$ |
\( T^{8} + \cdots - 81\!\cdots\!00 \)
|
| $67$ |
\( T^{8} + \cdots - 29\!\cdots\!08 \)
|
| $71$ |
\( T^{8} + \cdots + 12\!\cdots\!52 \)
|
| $73$ |
\( T^{8} + \cdots - 45\!\cdots\!76 \)
|
| $79$ |
\( T^{8} + \cdots + 77\!\cdots\!60 \)
|
| $83$ |
\( T^{8} + \cdots + 69\!\cdots\!04 \)
|
| $89$ |
\( T^{8} + \cdots - 37\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots - 42\!\cdots\!00 \)
|
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